|
It has been remarked that the continuous is effectually
distinguished from the discrete by their possessing the one a
common, the other a separate, limit.
The same principle gives rise to the numerical distinction
between odd and even; and it holds good that if there are
differentiae found in both contraries, they are either to be
abandoned to the objects numbered, or else to be considered as
differentiae of the abstract numbers, and not of the numbers
manifested in the sensible objects. If the numbers are logically
separable from the objects, that is no reason why we should not
think of them as sharing the same differentiae.
But how are we to differentiate the continuous, comprising as it
does line, surface and solid? The line may be rated as of one
dimension, the surface as of two dimensions, the solid as of
three, if we are only making a calculation and do not suppose
that we are dividing the continuous into its species; for it is
an invariable rule that numbers, thus grouped as prior and
posterior, cannot be brought into a common genus; there is no
common basis in first, second and third dimensions. Yet there is
a sense in which they would appear to be equal- namely, as pure
measures of Quantity: of higher and lower dimensions, they are
not however more or less quantitative.
Numbers have similarly a common property in their being numbers
all; and the truth may well be, not that One creates two, and two
creates three, but that all have a common source.
Suppose, however, that they are not derived from any source
whatever, but merely exist; we at any rate conceive them as being
derived, and so may be assumed to regard the smaller as taking
priority over the greater: yet, even so, by the mere fact of
their being numbers they are reducible to a single type.
What applies to numbers is equally true of magnitudes; though
here we have to distinguish between line, surface and solid- the
last also referred to as "body"- in the ground that, while all
are magnitudes, they differ specifically.
It remains to enquire whether these species are themselves to be
divided: the line into straight, circular, spiral; the surface
into rectilinear and circular figures; the solid into the various
solid figures- sphere and polyhedra: whether these last should be
subdivided, as by the geometers, into those contained by
triangular and quadrilateral planes: and whether a further
division of the latter should be performed.
|
|